Cubic critical, medium range

Average Error: 43.9 → 2.9

Time: 14.5s

Precision: binary64

Cost: 53568

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

↓

\[-0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + {c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5 (/ c b))
  (+
   (+
    (* a (* (/ (pow c 2.0) (pow b 3.0)) -0.375))
    (* (pow c 4.0) (* (/ (pow a 3.0) (pow b 7.0)) -1.0546875)))
   (* (pow a 2.0) (* (/ (pow c 3.0) (pow b 5.0)) -0.5625)))))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (((a * ((pow(c, 2.0) / pow(b, 3.0)) * -0.375)) + (pow(c, 4.0) * ((pow(a, 3.0) / pow(b, 7.0)) * -1.0546875))) + (pow(a, 2.0) * ((pow(c, 3.0) / pow(b, 5.0)) * -0.5625)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

↓

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + (((a * (((c ** 2.0d0) / (b ** 3.0d0)) * (-0.375d0))) + ((c ** 4.0d0) * (((a ** 3.0d0) / (b ** 7.0d0)) * (-1.0546875d0)))) + ((a ** 2.0d0) * (((c ** 3.0d0) / (b ** 5.0d0)) * (-0.5625d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

↓

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (((a * ((Math.pow(c, 2.0) / Math.pow(b, 3.0)) * -0.375)) + (Math.pow(c, 4.0) * ((Math.pow(a, 3.0) / Math.pow(b, 7.0)) * -1.0546875))) + (Math.pow(a, 2.0) * ((Math.pow(c, 3.0) / Math.pow(b, 5.0)) * -0.5625)));
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

↓

def code(a, b, c):
	return (-0.5 * (c / b)) + (((a * ((math.pow(c, 2.0) / math.pow(b, 3.0)) * -0.375)) + (math.pow(c, 4.0) * ((math.pow(a, 3.0) / math.pow(b, 7.0)) * -1.0546875))) + (math.pow(a, 2.0) * ((math.pow(c, 3.0) / math.pow(b, 5.0)) * -0.5625)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

↓

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(Float64(a * Float64(Float64((c ^ 2.0) / (b ^ 3.0)) * -0.375)) + Float64((c ^ 4.0) * Float64(Float64((a ^ 3.0) / (b ^ 7.0)) * -1.0546875))) + Float64((a ^ 2.0) * Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * -0.5625))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

↓

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (((a * (((c ^ 2.0) / (b ^ 3.0)) * -0.375)) + ((c ^ 4.0) * (((a ^ 3.0) / (b ^ 7.0)) * -1.0546875))) + ((a ^ 2.0) * (((c ^ 3.0) / (b ^ 5.0)) * -0.5625)));
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * N[(N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 4.0], $MachinePrecision] * N[(N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.9

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around inf 2.9

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

3. Simplified 2.9

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right)} \]
   Proof

   [Start] 2.9	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]
   rational.json-simplify-41 [=>] 2.9	\[ -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \color{blue}{\left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right)\right)} \]
   rational.json-simplify-41 [=>] 2.9	\[ \color{blue}{-0.5 \cdot \frac{c}{b} + \left(\left(-0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}\right) + -0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}\right)} \]

4. Taylor expanded in a around 0 2.9

   \[\leadsto -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \color{blue}{-0.16666666666666666 \cdot \frac{\left(5.0625 \cdot {c}^{4} + 1.265625 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{7}}}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]

5. Simplified 2.9

   \[\leadsto -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \color{blue}{-0.16666666666666666 \cdot \left({a}^{3} \cdot \frac{{c}^{4} \cdot 6.328125}{{b}^{7}}\right)}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   Proof

   [Start] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -0.16666666666666666 \cdot \frac{\left(5.0625 \cdot {c}^{4} + 1.265625 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{7}}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   rational.json-simplify-49 [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -0.16666666666666666 \cdot \color{blue}{\left({a}^{3} \cdot \frac{5.0625 \cdot {c}^{4} + 1.265625 \cdot {c}^{4}}{{b}^{7}}\right)}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   rational.json-simplify-2 [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -0.16666666666666666 \cdot \left({a}^{3} \cdot \frac{\color{blue}{{c}^{4} \cdot 5.0625} + 1.265625 \cdot {c}^{4}}{{b}^{7}}\right)\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   rational.json-simplify-51 [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -0.16666666666666666 \cdot \left({a}^{3} \cdot \frac{\color{blue}{{c}^{4} \cdot \left(1.265625 + 5.0625\right)}}{{b}^{7}}\right)\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   metadata-eval [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -0.16666666666666666 \cdot \left({a}^{3} \cdot \frac{{c}^{4} \cdot \color{blue}{6.328125}}{{b}^{7}}\right)\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]

6. Taylor expanded in a around 0 2.9

   \[\leadsto -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \color{blue}{-1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]

7. Simplified 2.9

   \[\leadsto -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \color{blue}{{c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   Proof

   [Start] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   rational.json-simplify-2 [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -1.0546875 \cdot \frac{\color{blue}{{a}^{3} \cdot {c}^{4}}}{{b}^{7}}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   rational.json-simplify-49 [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + -1.0546875 \cdot \color{blue}{\left({c}^{4} \cdot \frac{{a}^{3}}{{b}^{7}}\right)}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]
   rational.json-simplify-43 [=>] 2.9	\[ -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + \color{blue}{{c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)}\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]

8. Final simplification 2.9

   \[\leadsto -0.5 \cdot \frac{c}{b} + \left(\left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + {c}^{4} \cdot \left(\frac{{a}^{3}}{{b}^{7}} \cdot -1.0546875\right)\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right)\right) \]

Alternatives

Alternative 1
Error	3.9
Cost	33664

\[-0.5 \cdot \frac{c}{b} + \left(a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right) + {a}^{2} \cdot \left(-0.5625 \cdot \frac{{c}^{3}}{{b}^{5}}\right)\right) \]

Alternative 2
Error	3.9
Cost	33664

\[\left(-0.5 \cdot \frac{c}{b} + a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot -0.375\right)\right) + {a}^{2} \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot -0.5625\right) \]

Alternative 3
Error	4.0
Cost	14272

\[\frac{0.3333333333333333}{-0.6666666666666666 \cdot \frac{b}{c} + \left(0.5 \cdot \frac{a}{b} + c \cdot \left(0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)} \]

Alternative 4
Error	6.0
Cost	832

\[\frac{0.3333333333333333}{0.5 \cdot \frac{a}{b} + -0.6666666666666666 \cdot \frac{b}{c}} \]

Alternative 5
Error	12.0
Cost	320

\[-0.5 \cdot \frac{c}{b} \]

Reproduce

herbie shell --seed 2023065 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))